A Special class of Hypersurfaces parametrized by lines of curvature in R4

Abstract

In this paper we study hypersurfaces in R 4 parametrized by lines of curvature with three distinct principal curvatures and with Laplace invariants m ji = m ki = 0, m jik = 0 for i, j, k distinct fixed indices. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and three vector valued functions of one variable, this family includes a classe of Dupin hypersurfaces. Moreover, we show that these vector valued functions are invariant under inversions and homotheties.